%\gray
\section{Morphisms of sites}
%%%%%%%%% SGA3, Tamme and Artin
%%%%%%http://modular.fas.harvard.edu/sga/sga/4-1/4-1t_278.html
%%%%%%http://modular.fas.harvard.edu/sga/sga/4-1/4-1t_277.html
%%%%%%http://modular.fas.harvard.edu/sga/sga/4-1/4-1t_265.html
% SGA
% http://fr.wikipedia.org/wiki/Pr%C3%A9faisceau
%http://ncatlab.org/nlab/show/Nisnevich+site
%http://math.northwestern.edu/~hoyois/papers/nisnevich.pdf
%étale and nisnevich sheaves
\noindent Having in mind that sites generalise the concept of covering to categories rather than the category topological spaces, a natural question about the possibility of generalising the concept of continous maps arises. We know that having a continuous map between topological spaces induces an adjunction between the categories of the (pre-)sheaves on its domain and codomain, as recalled below. In this section we study the functors between sites, given a particular attention to sheave preserving functor in the sense of the induced push-forward and pull-back functors recalled below.\\
Recall that if $f:X\rightarrow Y$ a continuous map of topological spaces, then the inverse image $f^{-1}$ preserves coverings, the we have the functor
\[f_{\ast}:\Shv X\rightarrow \Shv Y\]
given on objects by $(f_{\ast}P)(V)=P(f^{-1}(V))$ for $V\subseteq Y$.

Then one may wants to define a functor $f^{\ast}:\Shv Y\rightarrow \Shv X$ such that $f^{\ast}(Q)(f^{-1}(V))=Q(V)$. Then for every $U\subseteq X$ open, and for every $V\subseteq Y$, open, with the inclusion, $U\subset f^{-1}(V)$, we should have the restriction $Q(V)=(f^{\ast}(Q))(f^{-1}(V))\rightarrow (f^{\ast}(Q))(U)$. The natural way to do so, is to try to define $f^{\ast}(Q)$ to be the universal such sheaf. So, it is defined by $(f^{\ast}(Q))(U)=\displaystyle\Colim_{U\subseteq f^{-1}(V)}Q(V) $, that any such sheave factorise through it, then it would be natural to define the $f^{\ast}$ using the coloimit, providing it defines a sheaf. Then, the restriction are the colimit injections. Then, one can easily show that $f^{\ast}(Q)$ is a sheaves, called the inverse image of $Q$ along $f$, and that $f^{\ast}$ is a left adjoint of $f_{\ast}$.


\noindent We will recall the contraction of the associated functors of pre-sheaves:\cite{Tam94}\\
\noindent Let $f^{-1}:\bcD\rightarrow\bcC$ be a functor, $\bcA^{\bcC}$, $\bcA^{\bcD}$ the categories of pre-sheaves with values in $\bcA$ (a cocomplet category, \tcr{does it have to satisfy other conditions?}). $f^{-1}$ is not necessary an inverse of any functor in any sense, the notation is just adopted to be in-line with the notation of the special case raised from continuous maps of topological spaces, described above.
Then there exist a functor
\[f_{\ast}:\bcA^{\bcC}\rightarrow \bcA^{\bcD}\]
given on objects by  $f_{\ast}(P)=P\circ f^{-1}$, and on morphisms by $f_{\ast}(\phi)$ is the morphism of pre-sheaves given by $f_{\ast}(\phi)_V=\phi_{ f^{-1}(V)}$.
\begin{proof}
\tcb{straightforward}
\end{proof}
\noindent Also, in analogue of the above discussion, there is the natural smallest functor
\[f^{\ast}_{pre}:\bcA^{\bcD}\rightarrow \bcA^{\bcC}\]
generated by the relations $f^{\ast}_{pre}(Q)(f^{-1}(V))=Q(V),\forall V\in \bcD, Q\in \bcA^{\bcD}$.\\
\tcr{What if there are $V,V'\in \bcD$ such that $f^{-1}(V)=f^{-1}(V')$, but $Q(V)\neq Q(V')$?} \\
$\forall U\in \bcC, V\in \bcD, U\rightarrow f^{-1}(V)$ in $\bcC$ we have the restriction $Q(V)\rightarrow f^{\ast}_{pre}(Q)(U)$, so we define $f^{\ast}_{pre}(Q)(U)$ to be the smallest such object in $\bcA$ that makes $f^{\ast}_{pre}(Q)$ into pre-sheaf. That is done formally as follows:\\
For every $U\in \bcC$, we define the category $\bcD_U^{f^{-1}}$ whose objects are pairs $(V,i_V)$, where $V\in \bcD$ and $i_V:U\rightarrow f^{-1}(V) $ in $\bcC$; and its morphisms are the morphisms $u:V\rightarrow V'$ in $\bcD$ that makes the following diagram commute:
\[\xymatrix{U\ar[rr]^{i_V}\ar[rrd]_{i_V}&&f^{-1}(V)\ar[d]^u\\
&&f^{-1}(V').
}
\]
We define the functor $i_U^{f^{-1}}:\bcD_U^{f^{-1}}\rightarrow \bcD$, given on objects by $i_U^{f^{-1}}(V,i_V)=V$. 

\noindent Then, for any $U\in \bcC$ we define $f^{\ast}_{pre}(Q)(U)=\displaystyle\Colim_{\bcD_U^{f^{-1}}}Q\circ i_U^{f^{-1}}$. For any morphism $\varphi:U'\rightarrow U$ in $\bcC$, we have the functor $\bcD_{\varphi}^{f^{-1}}:\bcD_U^{f^{-1}}\rightarrow \bcD_{U'}^{f^{-1}}$, that sends $(V,i_V)$ to $(V,i_V\circ \varphi)$, and keeps the morphisms unchanged. Then, it is easy to see that the following diagram commute:
\[
\xymatrix{
\bcD_U^{f^{-1}}\ar[rr]^{i_U^{f^{-1}}}\ar[d]_{\bcD_{\varphi}^{f^{-1}}}&&\bcD\\
\bcD_{U'}^{f^{-1}}\ar[rru]_{i_{U'}^{f^{-1}}}
}
\]
Hence, \tcr{there is a canonical morphism }
\[\displaystyle\Colim_{\bcD_U^{f^{-1}}}Q\circ i_U^{f^{-1}}\rightarrow \displaystyle\Colim_{\bcD_{U'}^{f^{-1}}}Q\circ i_{U'}^{f^{-1}}\]makes the colimit coins commute. Then, we define $f^{\ast}_{pre}(Q)(\varphi)$ to be that canonical morphism. One can easily see that $f^{\ast}_{pre}(Q)$ is pre-sheaf, for being defined by universal property on morphisms.\\
\tcr{prove the sub-statement in general}.

 One can easily see that $f^{\ast}_{pre}(Q)(f^{-1}(V))=Q(V)$, that $id_{f^{-1}(V)}$ is terminal in $\bcD_{f^{-1}(V)}^{f^{-1}}$. \tcb{type it's a functor}
\begin{lemma}{\em \cite[Th 2.3.1]{tam94}
Let $f^{-1}:\bcD\rightarrow\bcC$ be a functor, $\bcA$ an additive category, then:
\begin{enumerate}
\item $f^{\ast}_{pre}$ is a left adjoint of $f_{\ast}$.
\item $f_{\ast}$ is additive, exact and commutes with colimits. \tcb{SGA4}
\item $f^{\ast}_{pre}$ is additive, right exact and commutes with colimits.
\item If $f^{\ast}_{pre}$ is exact, then $f_{\ast}$ maps injective objects in $\bcA^{\bcD}$ to injective objects in $\bcA^{\bcC}$.
\item \tcr{Does $f^{\ast}_{pre}$ maps (projective) injective objects in $\bcA^{\bcC}$ to (projective) injective objects in $\bcA^{\bcD}$?}
\item $f^{\ast}_{pre}$ maps representable pre-sheaves $Hom(-,U), U\in \bcC$ to the representable pre-sheaves $Hom(-,f^{-1}(U))$.
\end{enumerate}}
\end{lemma}
\begin{proof}
\begin{enumerate}
\item Additive structure on $\bcA^{\bcD}$ and $\bcA^{\bcD}$ are given section-wise, then $\forall \phi, \phi':P\rightarrow P'$ in $\bcA^{\bcC}$ ,$V\in \bcD$ we have:
\[(f_{\ast}(\phi+\phi'))_V=(\phi+\phi')_{ f^{-1}(V)}=\phi_{ f^{-1}(V)}+\phi'_{ f^{-1}(V)}=(f_{\ast}(\phi))_V+(f_{\ast}(\phi))_V=(f_{\ast}(\phi)+f_{\ast}(\phi))_V\].
Hence, $f_{\ast}(\phi+\phi')=f_{\ast}(\phi)+f_{\ast}(\phi)$, and $f_{\ast}$ is additive.
\end{enumerate}
\tcb{type, also use artin}
\end{proof}

In the case of continuous maps topological spaces, the push-forward and pull-back functors map sheaves to sheaves, and that does not hold for any non necessary continuous map. That also holds in general as in the below example. We are interested in functors between underlying categories of sites that its push-forward or pull-back functors preserves sheaves, and studying their properties.
\begin{example}
Let $\bcC_{indes},\bcC_{can}$ be the indescrete and canonical sites of a category $\bcC$, let $f^{-1}:\bcC_{indes}\rightarrow\bcC_{can}$ be the identity functor, then it is readily seen that $f^{\ast}_{pre}$, and $f_{\ast}$ are the identity functor. 
\tcb{Consider the sheaf....}.
\end{example}
\begin{terminology}
%\tcb{\noindent In order to ease notion and keep in-line with Atrin's notaion, for the functor $f:\bcC\rightarrow \bcD$, we denote the push-forward functor $\bcA^{\bcD}\rightarrow \bcA^{\bcC}$ by $f^p$, and the pull-back $\bcA^{\bcC}\rightarrow \bcA^{\bcD}$ by $f_p$.}
\end{terminology}
\begin{definition}[Continuous maps of sites]
{\em \noindent A continuous map of sites $f:\bcC_{\tau},\bcD_{\tau'}$ is a functor $f^{-1}:\bcD\rightarrow\bcC$ be a functor, such that $f_{\ast}$ maps sheaves on $\bcC_{\tau}$ to sheaves on $\bcD_{\tau'}$.}
\end{definition}

\begin{lemma}
{\em Let $\bcC_{\tau},\bcD_{\tau'}$ be sites, $f^{-1}:\bcD\rightarrow\bcC$ be a functor. Then, $f^{-1}$ defines a continuous map of sites $f$ iff....}
\end{lemma}
\tcr{For the continuous map $f:\bcC_{\tau},\bcD_{\tau'}$, $f_{\ast}$ induced a functor $f_{\ast}:\Shv_{\tau}(\bcC)\rightarrow \Shv_{\tau'}(\bcD)$, that has a left adoint $f^{\ast}$ obtain by the composition $f^{\ast}=\#\circ f^{\ast}_{pre}\circ i$. Type it in general for any two adjunctions}
\begin{definition}[Morphisms of sites]
{\em A continuous map of sites $f:\bcC_{\tau},\bcD_{\tau'}$ is called a morphism of sites if $f^{\ast}$ commutes with finite limits.}
\end{definition}
\tcb{type implication of this property}.
\begin{lemma}
{\em \noindent Let $\bcC_{\tau},\bcD_{\tau'}$ be sites, $f^{-1}:\bcD\rightarrow\bcC$ be a functor of the underlying categories. Then $f^{-1}$ defines a morphism of sites $f:\bcC_{\tau}\rightarrow \bcD_{\tau'}$ if it satisfies the following conditions, for every $\bcV=\{g_j:V_j\rightarrow Y\}_{\{j\in J\}}\in Cov_{\tau' }(Y)$, and $V\rightarrow Y$ in $\bcD$:
\begin{itemize}
\item $\{f^{-1}(g_j)\}_{\{j\in J\}}\in Cov_{\tau}(f^{-1}(Y))$.
\item The canonical morphism $f^{-1}(V_j)\times_{f^{-1}(Y)}f^{-1}(V)\rightarrow f^{-1}(V_i\times_{Y}V)$ is an isomorphism, $\forall j\in J$.
\end{itemize}}
\end{lemma}

\begin{proof}
\noindent Assume the conditions of the lemma, we need to show that $f_{\ast}$ maps sheaves on $\bcC_{\tau}$ to sheaves on $\bcD_{\tau'}$, and that $f^{\ast}$ commutes with finite limits.\\
Let $P$ an $\bcA\!-\!$sheave on $\bcC_{\tau}$, $\bcV=\{g_j:V_j\rightarrow Y\}_{\{j\in J\}}\in Cov_{\tau'}(Y)$ for $Y\in \bcD$, then $\bcU:=\{f^{-1}(g_j)\}_{\{j\in J\}}\in Cov_{\tau}(f^{-1}(Y))$. Hence, the following diagram commute, that $P$ a sheave on $\bcC_{\tau}$:
\[
\xymatrix{ P(f^{-1}(Y))\rightarrow\displaystyle\prod_{j\in J}P(f^{-1}(V_j))\ar@<-3pt>[r]\ar@<3pt>[r]&\displaystyle\prod_{j,j'\in J}P(f^{-1}(V_j)\times_{f^{-1}(Y)}f^{-1}(V_{j'}))}
\]
Then, the definition of $f_{\ast}$, and the second condition of the lemma implies that the following diagram is exact, hence $f_{\ast}P$ is a sheaf on $\bcD_{\tau'}$.
\[
\xymatrix{ f_{\ast}P(Y)\rightarrow\displaystyle\prod_{j\in J}f_{\ast}P(V_j)\ar@<-3pt>[r]\ar@<3pt>[r]&\displaystyle\prod_{j,j'\in J}f_{\ast}P(V_j\times_Y V_{j'})}
\]
\tcr{type the rest}
\vspace{.4cm}
Now, assume that $f^{-1}$ defines a morphism of sites $f:\bcC_{\tau}\rightarrow \bcD_{\tau'}$, we need to show that the conditions hold:
\tcb{use adjointness}\\
Let $Y\in D$  $\bcV=\{g_j:V_j\rightarrow Y\}_{\{j\in J\}}\in Cov_{\tau' }(Y)$, and $V\rightarrow Y$ in $\bcD$:
\tcr{type}
\begin{itemize}
\item 
\item
\end{itemize}

\end{proof}
\begin{example}
\tcb{An example of a morphism of sites $f$, such that $f^{\ast}$ does not preserves sheaves.}
\end{example}

\begin{example}
The identity functor defines a continuous map of sites $f:\Sch/S_{\acute{e}t}\rightarrow f:\Sch/S_{Nis}$, that every \'{e}tale sheave is a Nisnevich one, because every Nisnevich covering is an \'{e}tale covering. \tcb{specify S}. (for pre-sheaves, $f_{\ast}$ and $f^{\ast}$ are the identity.
\tcb{attention to $F(\emptyset)$.} \tcr{Is it a morphism of sites?}
\end{example}
\noindent The contrary to the previous statement does not hold, as shown in this counter example:
\begin{counterexample}
\tcr{?}
\end{counterexample}
...................
%\begin{definition}[Cocontinuous maps of sites]{\em }\end{definition}